Integrable evolution equations on spaces of tensor densities and their peakon solutions
2010 (English)In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 299, no 1, 129-161 p.Article in journal (Refereed) Published
We study a family of equations defined on the space of tensor densities of weight λ on the circle and introduce two integrable PDE. One of the equations turns out to be closely related to the inviscid Burgers equation while the other has not been identified in any form before. We present their Lax pair formulations and describe their bihamiltonian structures. We prove local wellposedness of the corresponding Cauchy problem and include results on blow-up as well as global existence of solutions. Moreover, we construct "peakon" and "multi-peakon" solutions for all λ ≠ 0, 1, and "shock-peakons" for λ = 3. We argue that there is a natural geometric framework for these equations that includes other well-known integrable equations and which is based on V. Arnold's approach to Euler equations on Lie groups.
Place, publisher, year, edition, pages
2010. Vol. 299, no 1, 129-161 p.
Mathematics Physical Sciences
IdentifiersURN: urn:nbn:se:kth:diva-163817DOI: 10.1007/s00220-010-1069-9ISI: 000280808500004ScopusID: 2-s2.0-77955550646OAI: oai:DiVA.org:kth-163817DiVA: diva2:802376
QC 201504272015-04-132015-04-122015-04-27Bibliographically approved